{
  "id": "1502.01560",
  "version": "v1",
  "published": "2015-02-05T14:11:28.000Z",
  "updated": "2015-02-05T14:11:28.000Z",
  "title": "Mass minimizers and concentration for nonlinear Choquard equations in $\\R^N$",
  "authors": [
    "Hong yu Ye"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\\frac{1}{2}\\ds\\int_{\\R^N}|\\nabla u|^2+\\frac{1}{2}\\ds\\int_{\\R^N}V(x)|u|^2-\\frac{1}{2p}\\ds\\int_{\\R^N}(I_\\al*|u|^p)|u|^p $$ on $\\widetilde{S}(c)=\\{u\\in H^1(\\R^N)|\\ \\int_{\\R^N}V(x)|u|^2<+\\infty,\\ |u|_2=c,c>0\\},$ where $N\\geq1$ $\\al\\in(0,N)$, $\\frac{N+\\alpha}{N}\\leq p<\\frac{N+\\alpha}{(N-2)_+}$ and $I_\\al:\\R^N\\rightarrow\\R$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\\widetilde{S}(c)$ when $V(x)\\equiv0$ for all $\\frac{N+\\alpha}{N}\\leq p<\\frac{N+\\alpha}{(N-2)_+}$. For the mass critical case $p=\\frac{N+\\alpha+2}{N}$, we show that if $0\\leq V(x)\\in L_{loc}^{\\infty}(\\R^N)$ and $\\lim\\limits_{|x|\\rightarrow+\\infty}V(x)=+\\infty$, then mass minimizers exist only if $0<c<c_*=|Q|_2$ and concentrate at the flattest minimum of $V$ as $c$ approaches $c_*$ from below, where $Q$ is a groundstate solution of $-\\Delta u+u=(I_\\alpha*|u|^{\\frac{N+\\alpha+2}{N}})|u|^{\\frac{N+\\alpha+2}{N}-2}u$ in $\\R^N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-05T14:11:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear choquard equation",
      "mass minimizers",
      "concentration",
      "sharp existence results",
      "riesz potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150201560Y"
    }
  }
}